That's an interesting idea. There may be several reasons why whales breach. It might be a way to communicate with other whales. A big, dramatic splash is sure likely to get the attention of any other whales. It's loud, makes waves, and sound travels faster in air. But that doesn't mean that your dad is wrong, because the same behavior might have different causes.
Think about yawning. It may be something you do when you're sleepy, bored, nervous, or just because someone else yawned. You many even do it to communicate with a friend ("This is boring, let's go!").
Let's think about how a scientist could test the idea that breaching is a way to scratch. First we should probably think about whether it would be an effective way to scratch. There's some massive weight slapping the surface of the water, so that might do the job. Then we might think about whether a whale would need to scratch that way. Their fins are certainly too short to reach most of their bodies. There's usually nothing else around to scratch against. They can't rub themselves on a tree like a bear can. They can get parasites that attach themselves to their skin, so it makes sense that they might breach to use the force of hitting the water to remove parasites
How would you actually test the idea, though? First, you would need a large group of whales for testing. We call this having a large sample size. It's important because each individual is different. If you want to apply your findings to all whales, you need a large, random sample of all possible whales. Realistically, that may be impossible, but let's say you got access to 20 whales.
Then you need a controlled experiment. If you want to know whether whales breach more when they have an itch, you need to randomly assign whales to either get the itch treatment or not. The ones that don't get the itch treatment are your control group. They have to be treated exactly the same way except for the itch treatment. For example, if you decided to make them itchy by injecting a little mosquito venom, the control whales would get an injection too, it just wouldn't have mosquito venom in it. The venom is the independent variable that you are manipulating to get your answer. Of course, maybe whales don't get itchy from mosquito bites, in which case you would need something else as your treatment.
You then need to measure the dependent variable. In this case, it would be how many times the whales breach within a particular time. It is best if the people who do the counting don't know which whales are the treatment or control whales. Even when we don't mean to influence the results, our expectations (itchy whales will breach more) can influence what we see. Say a whale sort of breaches, but doesn't get all the way in the air. We need to be fair about whether we count that as a breach or not. If we don't know which whale is in which group, that can't influence our decision.
We might want to switch which whales get the itchy treatment or not.
Then we would analyze our data to see whether there was more breaching by itchy whales.
Can you think of other ways to make whales itchy? Why isn't it a fair test to just see which whales are breaching, then count how many skin parasites they have?
If you're interested in questions like this, you may want to study animal behavior.
Thanks for asking,